Surface area is the total area of the faces and curved surface of a solid figure.
Mathematical description of the surface area is considerably more involved than the definition of arc length or polyhedra (objects with flat polygonal faces) the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of the surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. General definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface. While areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of the area requires a lot of care. of a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface S is a union of finitely many pieces S1, , Sr which do not overlap except at their boundaries then
Common formulas
Surface areas of common solids Shape Cube Rectangular prism All Prisms Sphere Spherical lune Equation Variables s = side length = length, w = width, h = height B = the area of one base, P = the perimeter of one base, h = height r = radius of sphere r = radius of sphere, = dihedral angle r = radius of the circular base, h = height of the cylinder
Closed cylinder
Lateral surface area of acone
s = slant height of the cone, r = radius of the circular base, h = height of the cone s = slant height of the cone, r = radius of the circular base, h = height of the cone B = area of base, P = perimeter of base, L = slant height
Full surface area of a cone Pyramid
Ratio of surface areas of a sphere and cylinder of the same Radius and Volume
A cone, sphere and cylinder of radius r and height h.
The above formulas can be used to show that the surface area of a sphere andcylinder of the same radius and height are in the ratio 2 : 3, as follows. Let the radius be r and the height be h (which is 2r for the sphere).
The discovery of this ratio is credited to Archimedes.
[3]
The surface-area-to-volume ratio also called the surface-to-volume ratio and variously denoted sa/volor SA:V, is the amount of surface area per unit volume of an object or collection of objects. The surface-areato-volume ratio is measured in units of inverse distance. A cube with sides of length a will have a surface area of 6a2 and a volume of a3. The surface to volume ratio for a cube is thus shown as .
For a given shape, SA:V is inversely proportional to size. A cube 2 m on a side has a ratio of 3 m1, half that of a cube 1 m on a side. On the converse, preserving SA:V as size increases requires changing to a less compact shape.
Examples
Shape Length Area Volume SA/V ratio SA/V ratio for unit volume
Tetrahedron
side
7.21
Cube
side
Octahedron
side
5.72
Dodecahedron
side
5.31
Icosahedron
side
5.148
Sphere
radius
4.836
Example of Cubes of varying size Area of Face Total Surface Area 6 m2 24 m2 96 m2 216 m2 384 m2 Volume of Cube 1 m3 8 m3 64 m3 216 m3 512 m3 Surface Area to Volume Ratio 6.0 m1 (or m2m3) 3.0 m1 1.5 m1 1.0 m1 0.75 m1
Side
1 m 1 m2 2 m 4 m2 4 m 16 m2 6 m 36 m2 8 m 64 m2 12 m 20 m
144 m2
864 m2
1728 m3
0.5 m1
400 m2
2400 m2
8000 m3
0.3 m1
